本文研究以下加权积分系统： u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x - y ∣ λ d y , x ∈ R + n + 1 , v ( x ) = ∫ R + n + 1 y n + 1 β g ( u ( y ) , v ( y ) ) ∣ x - y ∣ λ d y , x ∈ R + n + 1 . left{begin{array}{l}uleft(x)=mathop{displaystyle int }limits_{{{mathbb{R}}}_{+}^{n+1}}frac{{y}_{n+1}^{beta }fleft(u(y),v(y))}{{| x-y| }^{lambda }}{rm{d}}y,hspace{1em}xin {{mathbb{R}}}_{+}^{n+1}},hspace{1.{{mathbb{R}}}_{+}^{n+1}中。 在 f f 和 g g 的性质结构条件下，我们用移动球的方法对正解进行分类。
Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.
{"title":"Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions","authors":"Jing Li, Wenchang Chu","doi":"10.1515/math-2024-0052","DOIUrl":"https://doi.org/10.1515/math-2024-0052","url":null,"abstract":"Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stability estimates are proposed for two variants of Markov processes defined by stochastic difference equations: uncontrolled and controlled. Processes of this type are widely used in applications where their “governing distributions” are known only approximately, for example, as statistical estimates obtained from real data. Therefore, the problem of estimating deviations of output characteristics arises. The Kantorovich metric is used to measure the variations of probability distributions that govern the processes. In the uncontrolled case, the Kantorovich distance between the stationary distributions of the initial process and its perturbation is evaluated. On the other hand, the control processes being compared are endowed with an expected total discounted cost, and the inequality for the corresponding stability index is obtained. The stability index measures the increase in costs when using the control policy optimal for the “approximating process.”
{"title":"Note on stability estimation of stochastic difference equations","authors":"Evgueni Gordienko, Juan Ruiz de Chavez","doi":"10.1515/math-2024-0041","DOIUrl":"https://doi.org/10.1515/math-2024-0041","url":null,"abstract":"Stability estimates are proposed for two variants of Markov processes defined by stochastic difference equations: uncontrolled and controlled. Processes of this type are widely used in applications where their “governing distributions” are known only approximately, for example, as statistical estimates obtained from real data. Therefore, the problem of estimating deviations of output characteristics arises. The Kantorovich metric is used to measure the variations of probability distributions that govern the processes. In the uncontrolled case, the Kantorovich distance between the stationary distributions of the initial process and its perturbation is evaluated. On the other hand, the control processes being compared are endowed with an expected total discounted cost, and the inequality for the corresponding stability index is obtained. The stability index measures the increase in costs when using the control policy optimal for the “approximating process.”","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size HH and the fine grid size hh satisfy h=O(H(2k+1)⁄(k+1))h={mathcal{O}}left({H}^{left(2k+1)/left(k+1)}) (k≥1kge 1), where kk is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
在本文中，我们提出了一种使用扩展混合有限元法求解二维双曲方程的方案。为了更高效地求解由此产生的非线性扩展混合有限元系统，我们提出了一种两步双网格算法。在粗网格和细网格上都证明了数值稳定性和误差估计。结果表明，只要粗网格尺寸 H H 和细网格尺寸 h h 满足 h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={mathcal{O}}left({H}^{left(2k+1)/left(k+1)}) ( k ≥ 1 kge 1 ) ，其中 k k 是主变量近似空间的度数，双网格法就能实现渐近最优近似。数值实验证明了所提方法的准确性和高效性。
{"title":"Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods","authors":"Keyan Wang","doi":"10.1515/math-2024-0048","DOIUrl":"https://doi.org/10.1515/math-2024-0048","url":null,"abstract":"In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>H</m:mi> </m:math> <jats:tex-math>H</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the fine grid size <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> </m:math> <jats:tex-math>h</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfy <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> <m:mo>=</m:mo> <m:mi mathvariant=\"script\">O</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁄</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>h={mathcal{O}}left({H}^{left(2k+1)/left(k+1)})</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>kge 1</jats:tex-math> </jats:alternatives> </jats:inline-formula>), where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0048_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> <jats:tex-math>k</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.
{"title":"Construction of a class of half-discrete Hilbert-type inequalities in the whole plane","authors":"Minghui You","doi":"10.1515/math-2024-0044","DOIUrl":"https://doi.org/10.1515/math-2024-0044","url":null,"abstract":"In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article surveys some results in the study of Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t→∞tto infty of the time-dependent solutions is proved.
本文概述了 Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145-191] 的研究中的一些结果，其中研究了粘性可压缩流体在二维域中受壁面加热影响的运动。给出了时变问题的全局存在性和唯一性定理，并在更严格的假设条件下给出了静止情况下的存在性和唯一性定理。证明了时变解在 t → ∞ tto infty 时的渐近行为定理。
{"title":"On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating","authors":"Angela Iannelli","doi":"10.1515/math-2024-0032","DOIUrl":"https://doi.org/10.1515/math-2024-0032","url":null,"abstract":"This article surveys some results in the study of Iannelli [<jats:italic>Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento</jats:italic>, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0032_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>t</m:mi> <m:mo>→</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>tto infty </jats:tex-math> </jats:alternatives> </jats:inline-formula> of the time-dependent solutions is proved.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Thabet Abdeljawad, Badreddine Meftah, Abdelghani Lakhdari, Manar A. Alqudah
This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.
{"title":"An extension of Schweitzer's inequality to Riemann-Liouville fractional integral","authors":"Thabet Abdeljawad, Badreddine Meftah, Abdelghani Lakhdari, Manar A. Alqudah","doi":"10.1515/math-2024-0043","DOIUrl":"https://doi.org/10.1515/math-2024-0043","url":null,"abstract":"This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.
{"title":"Strong laws for weighted sums of widely orthant dependent random variables and applications","authors":"Yong Zhu, Wei Wang, Kan Chen","doi":"10.1515/math-2024-0027","DOIUrl":"https://doi.org/10.1515/math-2024-0027","url":null,"abstract":"In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of generalized metric spaces is an active topic in general topology. In this article, we utilize the concepts of ideal convergence and networks to discuss the metrization problem and the mutual classification problem between spaces and mappings in topological spaces. We define ℐ{mathcal{ {mathcal I} }}-snsn-metrizable spaces, obtain several characterizations of ℐ{mathcal{ {mathcal I} }}-snsn-metrizable spaces, and establish some mapping relations between ℐ{mathcal{ {mathcal I} }}-snsn-metrizable spaces and semi-metric spaces. These not only generalize some theorems in generalized metric theory, but also find further applications of ideal convergence in general topology.
广义度量空间理论是广义拓扑学中一个活跃的话题。在本文中，我们利用理想收敛和网络的概念来讨论拓扑空间中的元化问题和空间与映射之间的相互分类问题。我们定义 ℐ {mathcal{ {mathcal I} }} 。}} - s n sn -metrizable 空间，得到ℐ {mathcal{ {mathcal I} }} 的几个特征。}} - s n sn 可三元空间，并在ℐ {mathcal{ {mathcal I} }} 之间建立了一些映射关系。}} - s n sn 可对称空间与半对称空间之间的映射关系。这些不仅概括了广义度量理论中的一些定理，而且发现了理想收敛在广义拓扑学中的进一步应用。
{"title":"ℐ-sn-metrizable spaces and the images of semi-metric spaces","authors":"Xiangeng Zhou, Fang Liu, Li Liu, Shou Lin","doi":"10.1515/math-2024-0053","DOIUrl":"https://doi.org/10.1515/math-2024-0053","url":null,"abstract":"The theory of generalized metric spaces is an active topic in general topology. In this article, we utilize the concepts of ideal convergence and networks to discuss the metrization problem and the mutual classification problem between spaces and mappings in topological spaces. We define <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐ</m:mi> </m:math> <jats:tex-math>{mathcal{ {mathcal I} }}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> <m:mi>n</m:mi> </m:math> <jats:tex-math>sn</jats:tex-math> </jats:alternatives> </jats:inline-formula>-metrizable spaces, obtain several characterizations of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐ</m:mi> </m:math> <jats:tex-math>{mathcal{ {mathcal I} }}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> <m:mi>n</m:mi> </m:math> <jats:tex-math>sn</jats:tex-math> </jats:alternatives> </jats:inline-formula>-metrizable spaces, and establish some mapping relations between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐ</m:mi> </m:math> <jats:tex-math>{mathcal{ {mathcal I} }}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0053_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>s</m:mi> <m:mi>n</m:mi> </m:math> <jats:tex-math>sn</jats:tex-math> </jats:alternatives> </jats:inline-formula>-metrizable spaces and semi-metric spaces. These not only generalize some theorems in generalized metric theory, but also find further applications of ideal convergence in general topology.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this study is to develop the theory of prime ideals in ordered semigroups. First, to ensure the existence of prime ideals, we study a class of ordered semigroups which will be denoted by SIP{{mathbb{S}}}_{IP}. And then we introduce the hull-kernel topology for the prime ideals P(S){mathcal{P}}left(S) and the topological properties like separation axioms, compactness and connectedness are studied. Finally, we focus on the subspace ℳ(S,I){mathcal{ {mathcal M} }}left(S,I), minimal prime ideals containing the ideal II in an ordered semigroup SS. We investigate topological properties of this subspace and connections between this subspace and the ordered semigroup SS.
本研究旨在发展有序半群中的素理想理论。首先，为了确保素理想的存在，我们研究了一类有序半群，用 S I P {{mathbb{S}}{{IP} 表示。然后，我们引入素理想 P ( S ) {mathcal{P}}left(S) 的赫尔核拓扑，并研究分离公理、紧凑性和连通性等拓扑性质。最后，我们聚焦于子空间ℳ ( S , I ) {mathcal{ {mathcal M}}left(S,I)}}left(S,I)，包含有序半群 S S 中理想 I I 的最小素理想。我们研究这个子空间的拓扑性质以及这个子空间与有序半群 S S 之间的联系。
{"title":"The hull-kernel topology on prime ideals in ordered semigroups","authors":"Huanrong Wu, Huarong Zhang","doi":"10.1515/math-2024-0050","DOIUrl":"https://doi.org/10.1515/math-2024-0050","url":null,"abstract":"The aim of this study is to develop the theory of prime ideals in ordered semigroups. First, to ensure the existence of prime ideals, we study a class of ordered semigroups which will be denoted by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi mathvariant=\"double-struck\">S</m:mi> </m:mrow> <m:mrow> <m:mi>I</m:mi> <m:mi>P</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{mathbb{S}}}_{IP}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. And then we introduce the hull-kernel topology for the prime ideals <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">P</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{mathcal{P}}left(S)</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the topological properties like separation axioms, compactness and connectedness are studied. Finally, we focus on the subspace <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℳ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{mathcal{ {mathcal M} }}left(S,I)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, minimal prime ideals containing the ideal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>I</m:mi> </m:math> <jats:tex-math>I</jats:tex-math> </jats:alternatives> </jats:inline-formula> in an ordered semigroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>S</m:mi> </m:math> <jats:tex-math>S</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We investigate topological properties of this subspace and connections between this subspace and the ordered semigroup <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0050_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>S</m:mi> </m:math> <jats:tex-math>S</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142191382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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